Bell's Theorem: Griffiths' Probability Density & Indeterministic QM

I'm sorry, I didn't see that before writing the summary. I'm assuming this is an exercise or practice problem and not a current conversation. Therefore, the summary is:In summary, Griffiths uses a parameter called \rho(\lambda) as the probability density for the hidden variable in his book "Introduction to Quantum Mechanics." This hidden variable is meant to make the theory deterministic, but it seems contradictory to use probability to describe it. Blumel's book provides a better explanation of Bell's Theorem and its connection to hidden variable theory. The idea is that each system has a unique property (\lambda) that determines its behavior, resulting in a statistical distribution for \lambda.
  • #1
touqra
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I am referring to Introduction to Quantum Mechanics (2nd Edition) by David J. Griffiths, page 425 on Bell's Theorem.

Griffiths used a parameter, called [tex]\rho(\lambda)[/tex] as the probability density for the hidden variable.

What I don't understand is that the hidden variable was suppose to make the theory deterministic, or specifically to show that quantum mechanics as an indeterministic theory is incomplete.
What is the reason that he can use probability to describe the hidden variable? Isn't this a contradiction?
 
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  • #3


This test of QM involves making many measurements on systems that have been "identically prepared". The idea of HV theory is that there might be some unknown property of the system (labeled by [itex]\lambda[/itex]) that determines exactly what will happen in that particular system, and that [itex]\lambda[/itex] varies from system to system. Thus the supposedly "identical preparation" of different systems is actually not identical, but depends in some way on the precise details and prior history of each system. This gives a statistical distribution for [itex]\lambda[/itex] which is called [itex]\rho(\lambda)[/itex].

EDIT: just noticed that the OP is from 2005!
 
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1. What is Bell's Theorem?

Bell's Theorem is a fundamental principle in quantum mechanics that addresses the issue of locality and determinism. It states that no physical theory can reproduce all the predictions of quantum mechanics unless it is non-local (i.e. it allows for instantaneous action at a distance) or indeterministic (i.e. it involves some randomness).

2. Who is Griffiths and what is his contribution to Bell's Theorem?

David J. Griffiths is a renowned physicist and author of the popular textbook "Introduction to Quantum Mechanics". In his 1984 paper, he demonstrated how Bell's Theorem can be derived from the principles of quantum mechanics, providing a clear and concise explanation of the theorem.

3. How does probability density play a role in Bell's Theorem?

In quantum mechanics, the probability of a particle's state is described by a probability density function. Bell's Theorem deals with the correlation between the measurements of two particles that were previously entangled. The probability density function allows us to calculate the probability of obtaining a certain result for each particle, which is essential in understanding the implications of Bell's Theorem.

4. Is Bell's Theorem widely accepted in the scientific community?

Yes, Bell's Theorem is widely accepted in the scientific community and has been tested and confirmed through numerous experiments. It has also been a topic of extensive research and discussion in the field of quantum mechanics, as it has important implications for our understanding of the nature of reality.

5. What are the implications of Bell's Theorem for our understanding of the universe?

Bell's Theorem challenges our traditional understanding of causality and determinism in the universe. It suggests that there may be hidden variables or factors at play in the quantum world that we are not yet aware of, and that the universe may not be as deterministic as we once thought. It also has implications for the concept of locality, as it suggests that there may be instantaneous connections between particles that are separated by great distances.

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